A finitely generated group that does not satisfy the generalized Burghelea Conjecture
aa r X i v : . [ m a t h . K T ] M a y A FINITELY GENERATED GROUP THAT DOES NOT SATISFY THEGENERALIZED BURGHELEA CONJECTURE
A. DRANISHNIKOV AND M. HULL
Abstract.
We construct a finitely generated group that does not satisfy the generalized Burgheleaconjecture. Generalized Burghelea Conjecture
In [2], Burghelea gave an explicit formula for the periodic cyclic homology of groups algebraswith rational coefficients (and more generally with coefficients in fields of characteristic zero):
P HC ∗ ( Q G ) = M [ x ] ∈h G i fin ,n ≥ H n + ∗ ( N x , Q ) ⊕ M [ x ] ∈h G i ∞ T ∗ ( x ; Q )where the group T ∗ ( x ; Q ) = lim ← { H ∗ +2 n ( N x , Q ) } .Here G x denoes the the centralizer of x in G , N x = G x / h x i is the reduced centralizer, h G i fin isthe set of conjugacy classes of elements of finite order, and h G i ∞ the set of conjugacy classes ofelements of infinite order. The bonding maps in the inverse sequences are the Gysin homomorphisms S : H m +2 ( N x , Q ) → H m ( N x , Q ) corresponding to the fibration B h x i ≃ S → BG x → BN x . Conjecture 1.1 (Generalized Burghelea Conjecture) . Let G be a discrete group, then T ∗ ( x ; Q ) = 0 for all x ∈ h G i ∞ . Burghelea stated the above conjecture for groups G which admit a finite K ( G,
1) [2]. In thesame paper Burghelea constructed a countable group that does not satisfy the Generalized Burghe-lea Conjecture. There are still no known counterexamples to the original version of Burgelea’sconjecture, and it is known to hold for many classes of groups [3, 4].The following is our main result.
Theorem 1.2.
There is a finitely generated group G that does not satisfy the Generalized BurgheleaConjecture. Our strategy is to show that the countable group constructed by Burghelea can be embeddedin a finitely generated group in a way that preserves centralizers. This embedding is based on thetheory of small cancellation over relatively hyperbolic groups developed by Osin [7].
Remark . Shortly after this paper was written the authors became aware that Engel andMarcinkowski also had constructions of finitely generated counterexamples to the generalizedBurghelea Conjecture, including a finitely presented counterexample and a counterexample of type F ∞ , using completely different methods. These constructions have since been added to [4]. Mathematics Subject Classification.
Key words and phrases.
Burghelea conjecture. Malnormal Embeddings
For a torsion-free group G hyperbolic relative to a subgroup H , a subgroup S of G is called suitable if S contains two infinite order elements f and g which are not conjugate to any elements of H andsuch that no non-trivial power of f is conjugate to a non-trivial power of g . This is equivalent to[7, Definition 2.2] since G is torsion-free. Indeed, the maximal virtually cylic subgroups containing f and g respectively are both cyclic, and since no power of f is equal to a power of g these cyclicsubgroups must intersect trivially.The following is a special case of [7, Theorem 2.4]. Theorem 2.1.
Let G be a torsion-free group hyperbolic relative to a subgroup H , let t ∈ G , and let S be a suitable subgroup of G . Then there exists a group G and an epimorphism γ : G → G suchthat:(1) γ | H is injective (so we identify H with its image in G ).(2) G is hyperbolic relative to H .(3) γ ( t ) ∈ γ ( S ) .(4) γ ( S ) is a suitable subgroup of G .(5) G is torsion-free. We inductively apply the previous theorem to construct the desired embedding. This can beextracted from the proof of [7, Theorem 2.6], but since it is not explicitly stated there we includethe proof below.Recall that a subgroup H of a group G is called malnormal if for all x ∈ G \ H , x − Hx ∩ H = { } . Theorem 2.2.
Let H be a torsion-free countable group. Then there exists a finitely generated group Γ which contains H as a malnormal subgroup.Proof. Let H = { h , h , h , ... } . We inductively define a sequence of quotients as follows: Let G = H ∗ F , where F = F ( x, y ) is the free group on { x, y } . Then G is torsion-free, hyperbolicrelative to H , and F is a suitable subgroup of G . Let α : G → G be the identity map. Supposenow we have constructed a torsion-free group G i together with an epimorphism α i : G → G i suchthat:(1) α i | H is injective (so we identify H with its image in G i )(2) G i is hyperbolic relative to H .(3) α i ( h j ) ∈ α i ( F ) for all 0 ≤ j ≤ i .(4) α i ( F ) is a suitable subgroup of G i .(5) G i is torsion-free.Given such G i , we can apply Theorem 2.1 to G i , H , t = h i +1 , and S = α i ( F ). Let G i +1 = G i be the quotient provided by Theorem 2.1. Define α i +1 = γ ◦ α i , where γ is the epimorphism givenby Theorem 2.1. Then Theorem 2.1 implies that α i : G i → G i +1 satisfies conditions (1)–(5).Let Γ be the direct limit of the sequence G → G → G ... , that is Γ = G / S ker( α i ). Let β : G → Γ be the natural quotient map. Note that β | F is surjective by construction. Indeed, G is generated by H ∪ { x, y } and for each h i ∈ H , α i ( h i ) ∈ α i ( F ), hence β ( h i ) ∈ β ( F ). Thus Γ isgenerated by { β ( x ) , β ( y ) } .Now β | H is injective, so H embeds in Γ; we identify H with its image in Γ. Suppose x ∈ Γ suchthat x − Hx ∩ H = { } . Then there exist g, h ∈ H \ { } such that x − gxh − = 1. Let ˜ x ∈ G such FINITELY GENERATED GROUP THAT DOES NOT SATISFY THE GENERALIZED BURGHELEA CONJECTURE3 that β (˜ x ) = x . Then for some i ≥
1, ˜ x − g ˜ xh − ∈ ker α i . This means that α i (˜ x ) − Hα i (˜ x ) ∩ H = { } .Since G i is hyperbolic relative to H , H is malnormal in G i by [7, Lemma 8.3b]. Hence α i (˜ x ) ∈ H ,which means that x = β (˜ x ) ∈ H . Therefore H is malnormal in Γ. (cid:3) Proof of Theorem 1.2
Proof of Theorem 1.2.
We start by reviewing the counterexample constructed by Burghelea. By theKan-Thurston theorem [1, 5], there exists a group G and a map t : K ( G, → C P ∞ which inducesan isomorphism on homology and cohomology. Burghelea observes that the group G can be chosento be torsion-free. The idea behind this observation is that since C P ∞ = S C P n , K ( G,
1) can beconstucted inductively as a union of the form K ( G,
1) = S K ( G i , K ( G i ,
1) = t − ( C P n )and each K ( G i ,
1) is a finite CW-complex (see, for example, the proof in [6] of the Kan-Thurstontheorem). Since G = S G i and each G i is torsion-free, G is also torsion-free.Note that H n ( C P ∞ ; Q ) ∼ = Q and the Gysin homomorphism S : H n +2 ( C P ∞ ; Q ) → H n ( C P ∞ ; Q ) for the canonical S -bundle S ∞ → C P ∞ is an isomorphism, hencelim ← { H n ( C P ∞ ; Q ) , S } ∼ = Q .Let 1 → Z = h x i → H → G → a ∈ H ∗ ( G ) = Z [ a ], deg ( a ) = 2.Note that H ∼ = π ( Y ), where Y is the pull-back of the bundle S ∞ → C P ∞ along t . Hence N x = H/ h x i ∼ = G , and T ( x ; Q ) ∼ = lim ← { H n ( G, Q ) } ∼ = lim ← { H n ( C P ∞ ; Q ) , S } ∼ = Q . H is thegroup constructed by Burghelea.We now apply Theorem 2.2 to obtain a malnormal embedding H → Γ into a finitely generatedgroup Γ, and we identify H with its image in Γ. Since H is malnormal, no elements of Γ \ H willcentralize x . Hence Γ x = H and N x ∼ = G . Then as before, we get that T ( x ; Q ) ∼ = Q = 0 in thegroup Γ. (cid:3) References [1] G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups,
J. Pure Appl. Algebra (1980), 1-47.[2] D. Burghelea, The cyclic homology of the group rings, Comment. Math. Helvetici (1985), 354-365.[3] A. Dranishnikov, On Burghelea Conjecture, arXiv:1612.08700.[4] A. Engel, M. Marcinkowski, Burghelea conjecture and asymptotic dimension of groups, Journal of Topologyand Analysis , https://doi.org/10.1142/S1793525319500559.[5] D. M. Kan and W. P. Thurston, Every connected space has the homology of a K ( π, Topology , (1976),253-258.[6] C.R.F. Maunder, A Short Proof of a Theorem of Kan and Thurston, Bull. London Math. Soc. (1981) (4):325-327.[7] D. Osin, Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. of Math.172